Continuous Piecewise Linear Elements Research Articles

A mixed FEM for the coupled Brinkman–Forchheimer/Darcy problem

This paper develops the a priori analysis of a mixed finite element method for the filtration of an incompressible fluid through a non-deformable saturated porous medium with heterogeneous permeability. Flows are governed by the Brinkman–Forchheimer and Darcy equations in the more and less permeable regions, respectively, and the corresponding transmission conditions are given by mass conservation and continuity of momentum. We consider the standard mixed formulation in the Brinkman–Forchheimer domain and the dual-mixed one in the Darcy region, and we impose the continuity of the normal velocities by introducing suitable Lagrange multiplier. The finite element discretization involves Bernardi–Raugel and Raviart–Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. Stability, convergence, and a priori error estimates for the associated Galerkin scheme are obtained. Numerical tests illustrate the theoretical results.

On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for the wave equation in the time domain, we consider a space-time boundary integral equation approach, based on its discretization by means of the coupling of a time discrete convolution quadrature of order 2, with the classical continuous piecewise linear boundary element method. The latter is either the Galerkin method or the collocation one, this being computationally much cheaper. For the efficient evaluation of most of the integrals generated by these two boundary element methods, but also for the subsequent analysis, we first define a new Gaussian quadrature. After recalling that stability and convergence have been proved for the Galerkin method, while for the collocation one we only have numerical evidences of these properties, to (partially) fill this gap, we first note that collocation can be obtained directly from the Galerkin system, after discretizing its inner product integral by using the composite trapezoidal rule, which in this case turns out to be the Gaussian rule mentioned above with $1$ node. Then, we show that for all (positive) time step-sizes $\varDelta _t$, when we let $h\rightarrow 0$, independently from $\varDelta _t$, the matrix of the linear system produced by the collocation method, properly normalized, converges, in the $\infty $-norm, to the corresponding Galerkin matrix. The behavior of the associated absolute and relative errors in the $\infty $-norm, as $h\rightarrow 0$, are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ and $O\big (h^{\frac {2}{3}}/|\!\ln \varDelta _t|+\varDelta _t^{-(1+\varepsilon )}h^{\frac {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln \varDelta _t|]\big )$, respectively, with $\varepsilon>0$ as close as one likes to $0$ and the $O(\cdot )$ constant independent from $\varDelta _t, h$. Some possible extensions of the investigation we carried out are also mentioned.

Recovery of a space-time-dependent diffusion coefficient in subdiffusion: stability, approximation and error analysis

Abstract In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order $\alpha \in (0,1)$ in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in $L^2(0,T;L^2(\varOmega ))$ under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal $L^p$ regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard $\ell ^2(L^2(\varOmega ))$ error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples.

Discretization of a Distributed Optimal Control Problem with a Stochastic Parabolic Equation Driven by Multiplicative Noise

A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $$ O(\tau ^{1/2} + h^2) $$ is rigorously derived.

A five-field augmented fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this work we introduce and analyze a new augmented fully-mixed formulation for the stationary Navier–Stokes/Darcy coupled problem. Our approach employs, on the free-fluid region, a technique previously applied to the stationary Navier–Stokes equations, which consists of the introduction of a modified pseudostress tensor involving the diffusive and convective terms, together with the pressure. In addition, by using the incompressibility condition, the pressure is eliminated, and since the convective term forces the free-fluid velocity to live in a smaller space than usual, we augment the resulting formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations. On the other hand, in the Darcy region we apply the usual dual-mixed formulation, which yields the introduction of the trace of the porous media pressure as an associated Lagrange multiplier. The latter is connected with the fact that one of the transmission conditions involving mass conservation becomes essential and must be imposed weakly. In this way, we obtain a five-field formulation where the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the aforementioned Lagrange multiplier, are the corresponding unknowns. The well-posedness analysis is carried out by combining the classical Babuška–Brezzi theory and the Banach fixed-point theorem. A proper adaptation of the arguments exploited in the continuous analysis allows us to state suitable hypotheses on the finite element subspaces ensuring that the associated Galerkin scheme is well-posed and convergent. In particular, Raviart–Thomas elements of lowest order for the pseudostress and the Darcy velocity, continuous piecewise linear polynomials for the free-fluid velocity, piecewise constants for the Darcy pressure, together with continuous piecewise linear elements for the Lagrange multiplier, constitute feasible choices. Finally, we provide several numerical results illustrating the performance of the Galerkin method and confirming the theoretical rates of convergence.

A discontinuous finite volume method for a coupled fracture model

In this paper, we present a discontinuous finite volume method for a hybrid-dimensional coupled fracture model on triangular meshes. In matrix and fracture, the pressure is approximated by the discontinuous and continuous piecewise linear elements, and in order to retain local conservation property, the velocity is sought in the discontinuous lowest-order Raviart–Thomas element space and piecewise linear element space. The proposed numerical scheme is only associated with the pressure, so we avoid solving an indefinite saddle-point problem. In addition, the discontinuous method eliminates the continuity of approximate functions across the interelement boundary, so it has the advantage of high localizability and easy handling of complicated geometries. Optimal order error estimates for the pressure and velocity are proved, and numerical experiments are carried out to confirm our theoretical analysis.

A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems

Abstract In this paper we introduce and analyze a fully mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of the lowest order for the stress of the fluid, Raviart–Thomas of the lowest order for the Darcy velocity, piecewise constants for the pressures and continuous piecewise linear elements for the vorticity. An a priori error estimates and associated rates of convergence are derived, and several numerical results illustrating the good performance of the method are reported.

A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data

SummaryWe analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L2‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L2‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.

Cut finite elements for convection in fractured domains

We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain, which is a union of manifolds of different dimensions such that a d dimensional component always resides on the boundary of a d+1 dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem is formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is posed on a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

RESIDUAL-BASED A POSTERIORI ERROR ESTIMATES FOR A CONFORMING FINITE ELEMENT DISCRETIZATION OF THE NAVIER-STOKES/DARCY COUPLED PROBLEM

We consider in this paper, a new a posteriori residual type error estimator of a conforming mixed finite element method for the coupling of fluid flow with porous media flow on isotropic meshes. Flows are governed by the Navier-Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. The finite element subspaces consider Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for a Lagrange multiplier defined on the interface. The posteriori error estimate is based on a suitable evaluation on the residual of the finite element solution. It is proven that the a posteriori error estimate provided in this paper is both reliable and efficient. In addition, our analysis can be extended to other finite element subspaces yielding a stable Galerkin scheme.

A posteriori error analysis of a fully-mixed formulation for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

In this paper we consider an augmented fully-mixed variational formulation that has been recently proposed for the coupling of the Navier–Stokes equations (with nonlinear viscosity) and the linear Darcy model, and derive a reliable and efficient residual-based a posteriori error estimator for the associated mixed finite element scheme. The finite element subspaces employed are piecewise constants, Raviart–Thomas elements of lowest order, continuous piecewise linear elements, and piecewise constants for the strain, Cauchy stress, velocity, and vorticity in the fluid, respectively, whereas Raviart–Thomas elements of lowest order for the velocity, piecewise constants for the pressure, and continuous piecewise linear elements for the traces, are considered in the porous medium. The proof of reliability of the estimator relies on a global inf–sup condition, suitable Helmholtz decompositions in the fluid and the porous medium, the local approximation properties of the Clément and Raviart–Thomas operators, and a smallness assumption on the data. In turn, inverse inequalities, the localization technique based on bubble functions, and known results from previous works, are the main tools yielding the efficiency estimate. Finally, several numerical results confirming the properties of the estimator and illustrating the performance of the associated adaptive algorithm are reported.

A conforming mixed finite element method for the Navier–Stokes/Darcy coupled problem

In this paper we develop the a priori analysis of a mixed finite element method for the coupling of fluid flow with porous media flow. Flows are governed by the Navier---Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers-Joseph-Saffman law. We consider the standard mixed formulation in the Navier---Stokes domain and the dual-mixed one in the Darcy region, which yields the introduction of the trace of the porous medium pressure as a suitable Lagrange multiplier. The finite element subspaces defining the discrete formulation employ Bernardi-Raugel and Raviart-Thomas elements for the velocities, piecewise constants for the pressures, and continuous piecewise linear elements for the Lagrange multiplier. We show stability, convergence, and a priori error estimates for the associated Galerkin scheme. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are reported.

A cut finite element method for coupled bulk-surface problems on time-dependent domains

In this contribution we present a new computational method for coupled bulk-surface problems on time-dependent domains. The method is based on a space–time formulation using discontinuous piecewise linear elements in time and continuous piecewise linear elements in space on a fixed background mesh. The domain is represented using a piecewise linear level set function on the background mesh and a cut finite element method is used to discretize the bulk and surface problems. In the cut finite element method the bilinear forms associated with the weak formulation of the problem are directly evaluated on the bulk domain and the surface defined by the level set, essentially using the restrictions of the piecewise linear functions to the computational domain. In addition a stabilization term is added to stabilize convection as well as the resulting algebraic system that is solved in each time step. We show in numerical examples that the resulting method is accurate and stable and results in well conditioned algebraic systems independent of the position of the interface relative to the background mesh.

A Priori Error Estimates for Three Dimensional Parabolic Optimal Control Problems with Pointwise Control

In this paper we provide an a priori error analysis for parabolic optimal control problems with a pointwise (Dirac-type) control in space on three-dimensional domains. The two-dimensional case was treated in [D. Leykekhman and B. Vexler, SIAM J. Numer. Anal., 51 (2013), pp. 2797--2821]; however, the three-dimensional case is technically much more involved. To approximate the problem we use standard continuous piecewise linear elements in space and the piecewise constant discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish ${\mathcal O}(\sqrt{k}+h)$ order of convergence rate for the control in the $L^2$ norm. This result improves almost twice the previously known estimate in [W. Gong, M. Hinze, and Z. Zhou, SIAM J. Control Optim., 52 (2014), pp. 97--119] and in addition does not require any relationship between the time step $k$ and the mesh size $h$. The main technical tools are discrete maximal parabolic regularity results and the best approximation-type estimate for the finite element error in the $L^\infty(\Omega;L^2(I))$ norm.

Finite element three-dimensional Stokes ice sheet dynamics model with enhanced local mass conservation

Parallel finite element nonlinear Stokes models have been successfully used for three-dimensional ice-sheet and glacier simulations due to their accuracy and efficiency, and their capability for easily handling highly irregular domains and different types of boundary conditions. In particular, the well-known Taylor–Hood element pair (continuous piecewise quadratic elements for velocity and continuous piecewise linear elements for pressure) results in highly accuracy velocity and pressure approximations. However, the Taylor–Hood element suffers from poor mass conservation which can lead to significant numerical mass balance errors for long-time simulations. In this paper, we develop and investigate a new finite element Stokes ice sheet dynamics model that enforces local element-wise mass conservation by enriching the pressure finite element space by adding the discontinuous piecewise constant pressure space to the Taylor–Hood pressure space. Through various numerical tests based on manufactured solutions, benchmark test problems, and the realistic Greenland ice-sheet, we demonstrate that, for ice-sheet modeling, the enriched Taylor–Hood finite element model remains highly accurate and efficient, and is physically more reliable and robust compared to the classic Taylor–Hood finite element model.

Analysis of an augmented fully-mixed approach for the coupling of quasi-Newtonian fluids and porous media

In this paper we introduce and analyze an augmented mixed finite element method for the coupling of quasi-Newtonian fluids and porous media. The flows are governed by a class of nonlinear Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and, in order to handle the nonlinearity involved in the Stokes region, we set the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Moreover, in order to facilitate the analysis, we augment the formulation in the fluid by incorporating a redundant Galerkin-type term arising from the quasi-Newtonian constitutive law multiplied by a suitable stabilization parameter. In this way, under a suitable and explicit choice of this parameter, a generalization of the Babuska–Brezzi theory is utilized to show the well-posedness of the continuous and discrete formulations and to derive the corresponding a priori error estimate. In particular, the feasible finite element subspaces include PEERS and Arnold–Falk–Winther elements for the stress, velocity and vorticity in the fluid, Raviart–Thomas elements and piecewise constants for the velocity and pressure in the porous medium, together with piecewise constant Stokes strain tensor and continuous piecewise linear elements for the traces. Next, we employ classical approaches, which include linearization techniques, Clément’s interpolator and Helmholtz’s decomposition, together with known efficiency estimates, to derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. Finally, several numerical results confirming the good performance of the method and the properties of the a posteriori error estimator, and illustrating the capability of the corresponding adaptive algorithm to identify the singular regions of the solution, are reported.

Numerical simulation of fluid–structure interaction of compressible flow and elastic structure

The paper is concerned with fluid–structure interaction problem of compressible flow and elastic structure in 2D domains with a special interest in medical applications to airflow in human vocal folds. The viscous flow in a time dependent domain is described by the Navier–Stokes equations written with the aid of the Arbitrary Lagrangian–Eulerian (ALE) method. The equations of motion for elastic deformations of the human vocal folds are coupled with the equations for the fluid flow using either loose or strong coupling. The space discretization of the flow problem is carried out by the discontinuous Galerkin finite element method. For the time discretization we use a semi-implicit scheme. In order to derive the space-time discretization of the elastic body problem, we apply the finite element method using continuous piecewise linear elements. For the time discretization we use the Newmark scheme. Results of numerical experiments are presented.

Analysis of a new stabilized finite element method for the reaction–convection–diffusion equations with a large reaction coefficient

In this paper, we propose and analyze a new stabilized finite element method using continuous piecewise linear (or bilinear) elements for solving 2D reaction–convection–diffusion equations. The equation under consideration involves a small diffusivity ε and a large reaction coefficient σ, leading to high Péclet number and high Damköhler number. In addition to giving error estimates of the approximations in L2 and H1 norms, we explicitly establish the dependence of error bounds on the diffusivity, the L∞ norm of convection field, the reaction coefficient and the mesh size. Our analysis shows that the proposed method is particularly suitable for problems with a small diffusivity and a large reaction coefficient, or more precisely, with a large mesh Péclet number and a large mesh Damköhler number. Several numerical examples exhibiting boundary or interior layers are given to illustrate the high accuracy and stability of the proposed method. The results obtained are also compared with those of existing stabilization methods.

Finite Element Analysis of an Optimal Control Problem in the Coefficients of Time-Harmonic Eddy Current Equations

This paper is concerned with an optimal control problem governed by time-harmonic eddy current equations on a Lipschitz polyhedral domain. The controls are given by scalar functions entering in the coefficients of the curl-curl differential operator in the state equation. We present a mathematical analysis of the optimal control problem, including sensitivity analysis, regularity results, existence of an optimal control, and optimality conditions. Based on these results, we study the finite element analysis of the optimal control problem. Here, the state is discretized by the lowest order edge elements of Nedelec’s first family, and the control is discretized by continuous piecewise linear elements. Our main findings are convergence results of the finite element discretization (without a rate).

A residual-based a posteriori error estimator for a fully-mixed formulation of the Stokes–Darcy coupled problem

In this paper we develop an a posteriori error analysis of a new fully mixed finite element method for the coupling of fluid flow with porous media flow in 2D. Flows are governed by the Stokes and Darcy equations, respectively, and the corresponding transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We consider dual-mixed formulations in both media, which yields the pseudostress and the velocity in the fluid, together with the velocity and the pressure in the porous medium, and the traces of the porous media pressure and the fluid velocity on the interface, as the resulting unknowns. The set of feasible finite element subspaces includes Raviart–Thomas elements of lowest order and piecewise constants for the velocities and pressures, respectively, in both domains, together with continuous piecewise linear elements for the traces. We derive a reliable and efficient residual-based a posteriori error estimator for the coupled problem. The proof of reliability makes use of the global inf–sup condition, Helmholtz decompositions in both media, and local approximation properties of the Clément interpolant and Raviart–Thomas operator. On the other hand, inverse inequalities, the localization technique based on element-bubble and edge-bubble functions, and known results from previous works, are the main tools for proving the efficiency of the estimator. Finally, some numerical results confirming the theoretical properties of this estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities of the solution, are reported.